Method and apparatus for reducing stick-slip

ABSTRACT

A method and apparatus for damping stick-slip oscillations in a drill string. In one embodiment a method includes damping the stick-slip oscillations using a drilling mechanism at the top of said drill string. The speed of rotation of the drilling mechanism is controlled using a PI controller. The control is characterized by tuning the PI controller so that the drilling mechanism absorbs most torsional energy from the drill string at a frequency that is at or near a frequency of the stick-slip oscillations.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a continuation under 35 U.S.C. § 120 of pending U.S.patent application Ser. No. 13/132,421 filed Jun. 2, 2011 entitled“Method and Apparatus for Reducing Stick-Slip,” which is a 35 U.S.C. §371 national phase application of PCT patent application numberPCT/GB2008/051144, filed 2 Dec. 2008, both of which are herebyincorporated herein by reference in their entireties.

FIELD OF THE INVENTION

The present invention relates to a method of damping stick-sliposcillations in a drill string, to a method of drilling a borehole, to amethod of estimating the instantaneous rotational speed of a bottom holeassembly, to a drilling mechanism for use in drilling a borehole, to anelectronic controller for use with a drilling mechanism, and to a methodof upgrading a drilling mechanism on a drilling rig.

BACKGROUND

Drilling an oil and/or gas well involves creation of a borehole ofconsiderable length, often up to several kilometers vertically and/orhorizontally by the time production begins. A drillstring comprises adrill bit at its lower end and lengths of drill pipe that are screwedtogether. The whole drillstring is turned by a drilling mechanism at thesurface, which in turn rotates the bit to extend the borehole. Thedrilling mechanism is typically a top drive or rotary table, each ofwhich is essentially a heavy flywheel connected to the top of thedrillstring.

The drillstring is an extremely slender structure relative to the lengthof the borehole, and during drilling the string is twisted several turnsbecause of torque-on-bit between about 500 and 10,000 Nm. Thedrillstring also displays a complicated dynamic behaviour comprisingaxial, lateral and torsional vibrations. Simultaneous measurements ofdrilling rotation at the surface and at the bit have revealed that thedrillstring often behaves as a torsional pendulum i.e. the top of thedrillstring rotates with a constant angular velocity, whereas the drillbit performs a rotation with varying angular velocity comprising aconstant part and a superimposed torsional vibration. In extreme cases,the torsional part becomes so large that the bit periodically comes to acomplete standstill, during which the drillstring is torqued-up untilthe bit suddenly rotates again at an angular velocity that is muchhigher than the angular velocity measured at the surface. Thisphenomenon is known as stick-slip.

Stick-slip has been studied for more than two decades and it isrecognized as a major source of problems, such as excessive bit wear,premature tool failures and poor drilling rate. One reason for this isthe high peak speeds occurring during in the slip phase. The highrotation speeds in turn lead to secondary effects like extreme axial andlateral accelerations and forces.

A large number of papers and articles have addressed the stick-slipproblem. Many papers focus on detecting stick-slip motion and oncontrolling the oscillations by operational means, such as addingfriction reducers to the mud, changing the rotation speed or the weighton bit. Even though these remedies sometimes help, they are eitherinsufficient or they represent a high extra costs.

A few papers also recommend applying smart control of the top drive todampen and prevent stick-slip oscillations. In IADC/SPE 18049 it wasdemonstrated that torque feed-back from a dedicated string torque sensorcould effectively cure stick-slip oscillations by adjusting the speed inresponse to the measured torque variations. In Jansen. J. D et al.“Active Damping of Self-Excited Torsional Vibrations in Oil WellDrillstrings”, 1995, Journal of Sound and Vibrations, 179(4), 647-668,it was suggested that the drawback of this approach is the need for anew and direct measurement of the string torque, which is not alreadyavailable. U.S. Pat. No. 5,117,926 disclosed that measurement as anothertype of feedback, based on the motor current (torque) and the speed.This system has been commercially available for many years under thetrade mark SOFT TORQUE®. The main disadvantage of this system is that itis a cascade control system using a torque feedback in series with thestiff speed controller. This increases the risk of instabilities atfrequencies higher than the stick-slip frequency.

IADC/SPE 28324 entitled “Application of High Sampling Rate DownholeMeasurements for Analysis and Cure of Stick-Slip in Drilling” disclosescontrol of a drilling process using driving equipment that includes aPID, a motor, a gear box and rotary table. The PID tries to maintain thedesired rotary speed of the drill string and it is suggested that thePID can be adjusted to prevent stick-slip. However, a simulation resultshows poor damping of stick-slip oscillations and it is concluded in thepaper that PID is too simple a servo-control system to preventstick-slip.

SUMMARY

Embodiments of the present disclosure are based on the insight that a PIor PID controller can in fact be used to obtain significant damping ofstick-slip oscillations by the drilling mechanism. In particular we haverealised that a PI or PID controller can be tuned to ensure efficientdamping torsional wave energy at and/or near the stick-slip frequency.

In contrast to some earlier systems, various embodiments disclosedherein are passive in the sense that neither string torque nor drivetorque is needed in a feed-back loop. Accordingly, damping can beachieved without the need for additional sensors to measure stringtorque, that otherwise increases complexity and cost.

According to some embodiments of the invention there is provided amethod of damping stick-slip oscillations in a drill string, whichmethod comprises the steps of:

(a) damping said stick-slip oscillations using a drilling mechanism atthe top of said drill string; and

(b) controlling the speed of rotation of said drilling mechanism using aPI controller; characterised by the step of

(c) tuning said PI controller so that said drilling mechanism absorbsmost torsional energy from said drill string at a frequency that is ator near a frequency of said stick-slip oscillations. The drillingmechanism may comprise a top drive or a rotary table for example. It isto be noted that the PI controller may be tuned once (for example uponencountering stick-slip for the first time, or in advance of drilling)and upon subsequent occurrences of stick-slip the PI controller may beused again without being re-tuned. Another possibility is for the PIcontroller to be re-tuned each time stick-slip is encountered, or evenperiodically during a stick-slip phase of drilling. In one embodiment,the PI controller is tuned before it is used to control the drillingmechanism to damp stick-slip oscillations. For example, the controllermay be tuned upon encountering stick-slip oscillations or it may beperformed periodically during drilling of the borehole as the drillstring length increases. One possibility is for the tuning to take placeas each 30 m section of drill pipe is added to the drill string.

In some embodiments said stick-slip oscillations comprise torsionalwaves propagating along said drill string, and step (c) comprisesadjusting an I-term of said PI controller to be dependent on anapproximate period of said stick slip oscillations and on the effectiveinertia of said drilling mechanism, whereby said drilling mechanism hasa frequency dependent reflection coefficient of said torsional waves,which reflection coefficient is substantially at a minimum at or nearsaid frequency of stick-slip oscillations. It is to be noted that it isnot essential for the peak absorption frequency of the drillingmechanism to match exactly the frequency of the stick-slip oscillations(which in some embodiments is the fundamental frequency). Due to the waythe PI controller is tuned, the drilling mechanism has a bandwidth offrequency absorption that is of a sufficient width (e.g. ˜0.4 Hz) andmagnitude (e.g. less than 85% reflection) so that damping is stilleffective even if the two frequencies are not exactly matched. Thisrepresents a significant advantage of the method. Typically, thefundamental frequency of stick-slip oscillations encountered in practicelies in the range 0.1 Hz (period 10 s) to 0.5 Hz (period 2 s) and thepeak absorption frequency caused by the PI controller may be within 50%of the fundamental frequency.

In some embodiments the lowest point of the frequency-reflectioncoefficient curve has a value between about 50% (0.5) and 90% (0.9). Ithas been found that reflection coefficients any higher than about 90%can make the drilling mechanism too “stiff” and reduce the chance ofsuccessfully damping the stick-slip oscillations. On the other hand, ithas been found that a reflection coefficient of any lower than about 50%makes the drilling mechanism too “soft” and drilling performance can beimpaired since the drilling mechanism responds to much smaller changesin drill string torque resulting in high speed variations.

The absorption bandwidth is inversely proportional to the effectiveinertia J of the drilling mechanism. Therefore as the effective inertiaof a drilling mechanism increases, it is preferable although notessential, that the approximate stick-slip period is estimated ormeasured more accurately to ensure that the frequency of greatestdamping is real stick-slip frequency.

In some embodiments, the method further comprises the step of adjustingsaid I-term according to I=ω_(s) ²J where ω_(s) is an approximate orestimated angular frequency of said stick-slip oscillations and J is theeffective inertia of said drilling mechanism. ω_(s) could of course beexpressed in terms of other parameters in this formula, such as theperiod or frequency.

In other embodiments, the method further comprises the step of measuringsaid approximate period of stick-slip oscillations for use in adjustingsaid I-term. In certain embodiments this measurement may be performedautomatically by a PLC for example. In that case, the approximate periodmay be determined using drill string geometry or it may be determined bycomputer observation of drive torque. Another possibility is for theapproximate period to be estimated by the driller, for example by timingwith a stop-watch torque oscillations shown on the driller's console, orby simply listening to changes in pitch of the motor(s) of the drillingmechanism and timing the period that way. The driller may input theapproximate stick-slip period into a console to be processed by a PLC totune the I-term of the PI controller.

In some embodiments, the method further comprises the step of adjustinga P-term of said PI controller to be the same order of magnitude as thecharacteristic impedance of said drillstring. In this way the reflectioncoefficient of the drilling mechanism can be reduced further, increasingthe damping effect.

In other embodiments, the method further comprises the step of adjustingsaid P-term such that said reflection coefficient does not vanishcompletely whereby a fundamental mode of said stick slip oscillations isinhibited from splitting into two new modes with different frequencies.

In some embodiments, the method further comprises the step of adjustingsaid P-term as P=ζ/a where a is a mobility factor that permitsadjustment of said P-term during drilling, whereby energy absorption ofsaid stick-slip oscillations by said drilling mechanism may be increasedor reduced. The mobility factor may be adjusted automatically by acontroller (e.g. PLC) and/or may be adjusted manually by the driller. Inthis way the softness of the drilling mechanism can be adjusted toachieve a balance between damping stick-slip oscillations and drillingperformance.

In some aspects the method further comprises the step of increasing saidmobility factor if the magnitude of said stick-slip oscillations do notsubstantially disappear or reduce. In this way the softness of thedrilling mechanism is increased (i.e. is made more responsive to smallertorque variations).

In other aspects the method further comprises the step of reducing saidmobility factor once the magnitude of said stick-slip oscillations hassubstantially disappeared or reduced, whereby drilling efficiency isincreased without re-appearance or increase in magnitude of saidstick-slip oscillations. In this way the softness of the drillingmechanism is reduced (i.e. is made less responsive to smaller torquevariations).

In some embodiments, said PI controller is separate from a drillingmechanism speed controller, the method further comprising the step ofbypassing said drilling mechanism speed controller with said PIcontroller during damping of said stick-slip oscillations. The PIcontroller may be provided on a drilling rig separate from the drillingmechanism, either on a new rig or as an upgrade to an existing rig inthe field. In use, when stick-slip oscillations occur, the PLC mayoverride the dedicated speed controller of the drilling mechanism(either automatically or under control of the driller) to control it asset out above.

In other embodiments, said drilling mechanism comprises said PIcontroller, the method further comprising the steps of tuning said PIcontroller when said stick-slip oscillations occur, and leaving said PIcontroller untuned otherwise. In such embodiments the PI controller maybe part of the dedicated speed controller in a drilling mechanism suchas a top drive. The PI controller may be provided as software installedon a PLC or other computer control mechanism at point of manufacture. Inuse, the PI controller is used continuously but may only need to betuned as described above when stick-slip oscillations occur. This tuningmay be activated automatically be remote drilling control software (e.g.a driller's console on or off site) and/or may be controlled by thedriller using a driller's console.

In some embodiments, the method further comprises the step of estimatingthe instantaneous rotational speed of a bottom hole assembly at thelower end of said drill string by combining a known torsional complianceof said drill string with variations in a drive torque of said drillingmechanism. This is a particularly useful optional feature of someembodiments of the invention and the output may be displayed on adriller's console or otherwise to help the driller to visualise what ishappening downhole.

In other embodiments, variations in drive torque are expressed only at afundamental frequency of said stick-slip oscillations, whereby saidestimating step is simplified such that it may be implemented by a PLCand performed in real time. The drive torque variations comprise afrequency spectrum which make the drive torque signal difficult toanalyse. We have realised that it is sufficient only to analyse thefundamental frequency component of the drive torque variations and thatthis enables the analysis to be performed in real-time on a PLC forexample.

In some embodiments, said estimating step comprises band pass filteringa drive torque signal with a band pass filter centred on an approximatefrequency of said stick-slip oscillations. This helps to remove most ofthe higher and lower frequencies in the torque signal. The approximatefrequency may be determined as described above.

In certain aspects, said estimate of instantaneous rotational speedcomprises determining a downhole speed using a total static drill stringcompliance and a phase parameter, and determining the sum of (i) a lowpass filtered signal representing a speed of rotation of said drillingmechanism and (ii) said downhole speed.

In other embodiments, the method further comprises the step ofdetermining said estimate periodically and outputting said estimate on adriller's console whereby a driller is provided with a substantiallyreal-time estimate of the instantaneous rotational speed of said bottomhole assembly.

In some embodiments, the method further comprises the step ofdetermining a stick-slip severity as the ratio of dynamic downhole speedamplitude over the mean rotational speed of said drilling mechanism,which stick-slip severity is useable to provide an output signalindicating the severity of stick-slip at that point in time.

According to some embodiments of the invention there is provided amethod of drilling a borehole, which method comprises the steps of:

(a) rotating a drill string with a drilling mechanism so as to rotate adrill bit at a lower end of said drill string whereby the earth'ssurface is penetrated; and

(b) in response to detection of stick-slip oscillations of said drillstring using a PI controller to control said drilling mechanism, whichPI controller has been tuned by a method according to any of claims 1 to11. It is to be noted that the PI controller may be tuned once (forexample upon encountering stick-slip for the first time) and uponsubsequent occurrences of stick-slip the PI controller may be usedwithout re-tuning. Of course, another possibility is for the PIcontroller to be re-tuned each time stick-slip is encountered, or evenas stick-slip is ongoing. The PI tuning method may therefore be usedselectively during drilling to counter stick-slip oscillations. At othertimes the PI controller may be left untuned so that a speed controllerof the drilling mechanism has a standard stiff behaviour (i.e. with areflection coefficient approximately equal to 1).

According to yet another embodiment of the invention there is provided amethod of estimating the instantaneous rotational speed of a bottom holeassembly at the lower end of a drill string, which method comprises thesteps of combining a known torsional compliance of said drill stringwith variations in a drive torque of said drilling mechanism. Such amethod may be performed either on or off site, either during drilling orafter drilling a section of the borehole. Such a method provides adrilling analysis tool to determine if the PI controller tuning aspectof embodiments of the invention would improve drilling performance.Accordingly, software to perform this method may be provided separatelyfrom software to perform the tuning method. The rotational speedestimating software may be provided in the controller of a new drillingmechanism (i.e. included a point of manufacture), as an upgrade to anexisting drilling mechanism (e.g. performed either on site or remotelyusing a satellite connection to a computer system on the drilling rig),or as a computer program product (e.g. on a CD-ROM or as a download froma web site) for installation by the rig operator.

In certain aspects, the rotational speed estimating method furthercomprises the estimating steps as set out above.

According to some embodiments of the invention there is provided adrilling mechanism for use in drilling a borehole, which drillingmechanism comprises an electronic controller having a PI controller andmemory storing computer executable instructions that when executed causesaid electronic controller to tune said PI controller according to thetuning steps set out above.

According to yet other embodiments of the invention there is provided anelectronic controller for use with a drilling mechanism for drilling aborehole, which electronic controller comprises a PI controller andmemory storing computer executable instructions that when executed causesaid electronic controller to tune said PI controller according to thetuning steps set out above. Such an electronic controller is useful forupgrading existing drilling rigs or where it is desirable or necessarythat the electronic controller is separate from the drilling mechanism.

According to further embodiments of the invention there is provided amethod of upgrading a drilling mechanism on a drilling rig, which methodcomprises the steps of uploading computer executable instructions to anelectronic controller on said drilling rig, which electronic controlleris for controlling operation of said drilling mechanism, wherein saidcomputer executable instructions comprise instructions for performing atuning method as set out above. Such an upgrade may be performed onsite, or may be performed remotely using a satellite connection forexample.

Certain embodiments of this invention are not limited to any particularindividual feature disclosed here, but include combinations of themdistinguished from the prior art in their structures, functions, and/orresults achieved. Features of various embodiments of the invention havebeen broadly described so that the detailed descriptions that follow maybe better understood, and in order that the contributions of thisinvention to the arts may be better appreciated. There are, of course,additional aspects of the various embodiments of the invention describedbelow and which may be included in the subject matter of the claims.Those skilled in the art who have the benefit of this disclosure, itsteachings, and suggestions will appreciate that the conceptions of thisdisclosure may be used as a creative basis for designing otherstructures, methods and systems for carrying out and practicing thepresent invention. The claims of this disclosure are to be read toinclude any legally equivalent devices or methods which do not departfrom the spirit and scope of the embodiments disclosed herein.

The present disclosure recognizes and addresses the previously mentionedproblems and long felt needs and provides a solution to those problemsand a satisfactory meeting of those needs in its various possibleembodiments and equivalents thereof. To one of skill in this art who hasthe benefits of this disclosure's realizations, teachings, andsuggestions, other purposes and advantages will be appreciated from thefollowing description of certain preferred embodiments, given for thepurpose of disclosure, when taken in conjunction with the accompanyingdrawings. The detail in these descriptions is not intended to thwartthis patent's object to claim this invention no matter how others maylater disguise it by variations in form, changes, or additions offurther improvements.

It will be understood that the various embodiments of the presentinvention may include one, some, or all of the disclosed, described,and/or enumerated improvements and/or technical advantages and/orelements in the claims.

BRIEF DESCRIPTION OF THE FIGURES

For a better understanding of exemplary embodiments of the invention,reference will now be made, by way of example only, to the accompanyingdrawings in which:

FIG. 1 is a schematic side view of a drilling rig using a methodaccording to various embodiments of the present invention;

FIG. 2 is a schematic block diagram of a PLC comprising a speedcontroller according to various embodiments of the present invention;

FIG. 3 is a graph of frequency versus reflection coefficient showing acomparison between a drilling mechanism using a speed controlleraccording to various embodiments of the present invention and a standardspeed controller;

FIGS. 4A′ and 4A″ is a screenshot of a first window available on adriller's console for configuring and controlling a method according tovarious embodiments of the present invention;

FIGS. 4B′ and 4B″ is a screenshot of a second window available on adriller's console that illustrates real-time drive torque and anestimate of downhole rotation speed of the bottom hole assembly in FIG.1;

FIGS. 5 and 6 are graphs illustrating results of a computer simulationmodelling a method according to various embodiments of the presentinvention; and

FIGS. 7 and 8 are graphs illustrating results of a test of a methodaccording to various embodiments of the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Referring to FIG. 1 a drilling rig 10 controls a drilling operationusing a drillstring 12 that comprises lengths of drill pipe 14 screwedtogether end to end. The drilling rig 10 may be any sort of oilfield,utility, mining or geothermal drilling rig, including: floating and landrigs, mobile and slant rigs, submersible, semi-submersible, platform,jack-up and drill ship. A typical drillstring is between 0 and 5 km inlength and has at its lowest part a number of drill collars or heavyweight drill pipe (HWDP). Drill collars are thicker-walled than drillpipe in order to resist buckling under the compression forces: drillpipe may have an outer diameter of 127 mm and a wall thickness of 9 mm,whereas drill collar may have an outer diameter of up to 250 mm and awall thickness of 85 mm for example.

A bottom hole assembly (BHA) 16 is positioned at the lower end of thedrillstring 12. A typical BHA 16 comprises a MWD transmitter 18 (whichmay be for example a wireline telemetry system, a mud pulse telemetrysystem, an electromagnetic telemetry system, an acoustic telemetrysystem, or a wired pipe telemetry system), centralisers 20, adirectional tool 22 (which can be sonde or collar mounted), stabilisers(fixed or variable) and a drill bit 28, which in use is rotated by a topdrive 30 via the drillstring 12.

The drilling rig 10 comprises a drilling mechanism 30. The function ofthe drilling mechanism 30 is to rotate the drill string 12 and therebythe drill 28 at the lower end. Presently most drilling rigs use topdrives to rotate the drillstring 12 and bit 28 to effect drilling.However, some drilling rigs use a rotary table and embodiments of theinvention are equally applicable to such rigs. Embodiments of theinvention are also equally useful in drilling any kind of borehole e.g.straight, deviated, horizontal or vertical.

A pump 32 is located at the surface and, in use, pumps drilling fluidthrough the drillstring 12 through the drill bit 28 and serves to cooland lubricate the bit during drilling, and to return cuttings to thesurface in the annulus formed between the drillstring and the wellbore(not shown).

Drilling data and information is displayed on a driller's console 34that comprises a touch screen 36 and user control apparatus e.g.keyboard (not shown) for controlling at least some of the drillingprocess. A digital PLC 38 sends and receives data to and from theconsole 34 and the top drive 30. In particular, a driller is able to seta speed command and a torque limit for the top drive to control thespeed at which the drill bit 28 rotates.

Referring to FIG. 2 the PLC 38 comprises a non-volatile flash memory 40(or other memory, such as a battery backed-up RAM). The memory storescomputer executable instructions that, when executed, perform thefunction of a speed controller 42 for the top drive 30. The speedcontroller 42 comprises a PI controller with anti-windup that functionsas described in greater detail below. In this embodiment the speedcontroller 42 is separate and distinct from the top drive 30. However,it is possible for the functionality of the speed controller asdescribed herein to be provided as part of the in-built dedicated speedcontroller of a top drive. Such in-built functionality may either beprovided at point of manufacture or may be part of a software upgradeperformed on a top drive, either on or off site. In other embodimentsthe PLC may be an analogue PLC.

PI Controller Tuning

The drill string 12 can be regarded as a transmission line for torsionalwaves. A variation of the friction torque at the drill bit 28 orelsewhere along the string generates a torsional wave that is propagatedupwards and is partially reflected at geometric discontinuities. Whenthe transmitted wave reaches the top drive 30, it is partially reflectedback into the drill string 12. For a top drive with a high inertiaand/or a stiff speed controller the reflection is nearly total so thatthat very little energy is absorbed by the top drive.

To quantify the top drive induced damping a complex reflectioncoefficient r for torsional waves at the drill string/top driveinterface may be defined as follows:

$\begin{matrix}{r = \frac{\zeta - Z}{\zeta + Z}} & (1)\end{matrix}$where ζ is the characteristic impedance for torsional waves and Z is theimpedance of the top drive. The characteristic impedance is proportionalto the cross sectional polar moment of inertia for the pipe, and variesroughly as the 4^(th) power of the pipe diameter. Note that thereflection coefficient is a complex function where, in general, both themagnitude and phase vary with frequency. If the speed control is stiff(i.e. |Z|>>ζ) then the reflection coefficient approaches −1 and nearly100% of the torsional wave energy is reflected back down the drillstring 12 by the top drive 30.

A complex representation of the top drive impedance may be derived asfollows. If the anti wind-up of the speed controller is neglected (whichis a non-linear function that limits torque) the drive torque of the topdrive 30 can be written as:T _(d) =P(Ω_(set)−Ω)+I∫(Ω_(set)−Ω)dt  (2)where P and I are respective the proportional and integration factors ofthe speed controller, and Ω is the actual output drive speed (in rad/s)and Ω_(set) is the set point of the drive speed (in rad/s). The drivetorque is actually the sum of motor torques times the gear ratio n_(g)(motor speed/output speed, >1). Notice that speed control here refers tothe output axis of the top drive. It is more common for the speedcontrol to refer to the motor axis; in that case the corresponding P andI values for the motor speed control would then be a factor 1/n_(g) ²lower than above.

Neglecting transmission losses, the equation of motion of the top driveoutput shaft is:

$\begin{matrix}{{J\frac{d\;\Omega}{d\; t}} = {T_{d} - T}} & (3)\end{matrix}$where J is the effective top drive inertia (including gear and drivemotors) and T is the external torque from the string. Combiningequations (2) and (3) and applying the Fourier transform gives thefollowing equation of motion:

$\begin{matrix}{{\left( {{i\;\omega\; J} + P + \frac{I}{i\;\omega}} \right)\Omega} = {{\left( {P + \frac{I}{i\;\omega}} \right)\Omega_{set}} - T}} & (4)\end{matrix}$

For simplicity, the same variable names have been used as in the timebased equations, although Ω, Ω_(set) and T now represent complexamplitudes. The implied time factor is exp(iωt), where i=√{square rootover (−1)} is the imaginary unit and ω=2πf is the angular frequency ofthe top drive 30. If we assume there is no cascade feedback through theset speed (as found in torque feed-back systems), the set speedamplitude vanishes and the equation above simplifies to:

$\begin{matrix}{T = {{- \left( {{i\;\omega\; J} + P + \frac{I}{i\;\omega}} \right)}\Omega}} & (5)\end{matrix}$The negative ratio −T/Ω is called the top end impedance Z of the string:

$\begin{matrix}{Z = {{i\;\omega\; J} + P + \frac{I}{i\;\omega}}} & (6)\end{matrix}$

This impedance can easily be generalized to an ideal PID controller, byadding a new term iωD to it, where D is the derivative term of thecontroller. A (normal) positive D-term will increase the effectiveinertia of the top drive while a negative factor will reduce it. Inpractice, because time differentiation of the measured speed is a noisedriving process that enhances the high frequency noise, the D-term in aPID controller is normally combined with a low pass filter. This filterintroduces a phase shift that makes the effective impedance morecomplicated and it therefore increases the risk of making instabilitiesat some frequencies, as explained below. Therefore, although a PIDcontroller with a D-term could be used to perform the tuning aspect ofsome embodiments of the invention, it is not recommended.

Combining equations (1) and (6) gives the following expression for thereflection coefficient, valid for PI type speed controlled top drives:

$\begin{matrix}{r = {- \frac{P - \zeta + {i \cdot \left( {{\omega\; J} - \frac{I}{\omega}} \right)}}{P + \zeta + {i \cdot \left( {{\omega\; J} - \frac{I}{\omega}} \right)}}}} & (7)\end{matrix}$Its magnitude has a minimum equal to:

$\begin{matrix}{{r}_{\min} = \frac{{P - \zeta}}{P + \zeta}} & (8)\end{matrix}$when the imaginary terms vanish, that is, when the angular frequency ofthe top drive 30 equals ω=√{square root over (I/J)}. For standard stiffspeed controllers this frequency is normally higher than the stick-slipfrequency (see FIG. 3 and associated description). However, we havediscovered that adjustment of the I-term of the PI controller alsoadjusts the peak absorption frequency of torsional waves by the topdrive 30. In particular, the I-term can be adjusted so that the maximumenergy absorption of torsional waves occurs at or near the stick-slipfrequency ω_(s) (i.e. when the magnitude of the reflection coefficientis minimum) as follows:I=ω _(s) ² J  (9)

This realization is significant since, as a first step to achieving gooddamping, the I-term of the PI controller is only dependent on thestick-slip frequency and the effective inertia of the top drive 30.Since the effective inertia is readily determined either in advance ofoperation or from figures quoted by the manufacturer, and since thestick-slip frequency can be readily determined during drilling, thismakes tuning of the PI controller straightforward whilst achieving goodenergy absorption by the top drive 30 of the stick-slip oscillations.

This first step in tuning the speed controller is a good first steptowards effective dampening of stick-slip oscillations. However, thedamping can be further improved. In particular the untuned P-term of thespeed controller is still too high, that is P>>ζ keeping the reflectioncoefficient close to −1. We have discovered that to obtain sufficientdamping of the stick-slip oscillations the P-term of the speedcontroller must be lowered so that it is of the same order of magnitudeas the characteristic impedance ζ. However, we have also discovered thatit is not desirable that the reflection coefficient vanishes completely,because that would radically change the dynamics of the drill string 12and the pendulum mode would split into two new modes, each with adifferent frequency. Furthermore an extremely soft speed controller thatabsorbs nearly all of the incident wave energy will cause very highspeed fluctuations of the top drive 30, in response to variations of thedownhole torque. This can reduce drilling efficiency.

We have discovered that the P-term can be selected as a non-integermultiple of the characteristic impedance ζ of the drill string, whichmay be expressed as P=ζ/a where a is a normalised mobility factor(dimensionless) less than unity, which is operator or computeradjustable within certain limits as described below. Having set theI-term to cause the imaginary part of equation (7) to vanish, settingthe P-term as described causes the minimum of the reflection coefficient(i.e. the peak absorption of energy by the top drive) at the stick-slipfrequency ω_(s) to become:

$\begin{matrix}{{r}_{\min} = \frac{1 - a}{1 + a}} & (10)\end{matrix}$

By permitting adjustment of the mobility factor a, the amount of energyreflected back down the drill string 12 can be controlled, withinlimits. These limits can be set by permitting only a certain range ofvalues for a, such as 0.05 to 0.33. This corresponds to a range for themagnitude of r_(min) from about 0.9 to 0.5. It is believed that thisrange enables the damping to be controlled so that stick-sliposcillations can be inhibited. If the speed controller 42 is muchstiffer than this (i.e. a reflection coefficient greater than about 0.9)we have found that too much of the torsional energy of the stick-sliposcillations is reflected back down the drill-string 12. Furthermore, ifthe speed controller 42 is too soft (i.e. a reflection coefficient lessthan about 0.5) we have found that drilling performance (e.g. in termsof ROP) can be affected.

A standard speed controller is designed to keep the motor speed constantand the true P and I constants refer to the motor axis. A typical drivemotor with a nominal power of 900 kW and a rotor inertia of J_(m)=25kgm² is typically controlled by a motor speed controller of P_(m)=500Nms. The speed controller I-factor is most often given indirectly as theP-factor divided by a time integration constant of typically τ_(i)=0.3s. As an example, assume a drive with one motor connected to the outputshaft with a gear having an inertia J_(g)=250 kgm² and a gear ratio ofn_(g)=5.32. The effective drive inertia is then J_(d)=J_(g)+n_(g)²J_(m)=960 kgm². The effective speed controller factors referred to theoutput shaft is similarly P=n_(g) ²P_(m)≈14000 Nms and I=P/τ_(i)≈47000Nm. In comparison, the characteristic impedance for a typical 5 inchpipe ζ≈340 Nms, which is only 2.4% of the real part of the driveimpedance.

FIG. 3 is a graph 48 of the magnitude of the reflection coefficient |r|versus frequency and shows the difference between a standard stiff speedcontroller (curve 50) and a speed controller tuned according to variousembodiments of the invention (curve 52). The latter is calculated with amobility factor of a=0.25 and an I-term providing maximum damping at 0.2Hz (5 s stick-slip period). At this frequency the reflection is reducedfrom about 0.993 (standard PI controller) to 0.6 (PI controller tuned asabove), which represents a dramatic improvement in the damping by thetop drive at the stick-slip frequency.

It is worth emphasizing the fact that in both cases the reflectioncoefficient stays below 1 but approaches this limit as the frequencyapproaches either zero or infinity. Therefore, the standardPI-controller never provides a negative damping that would otherwiseamplify torsional vibration components. However, the damping is poor faraway from the relatively narrow the absorption band at 1-2 Hz. Incontrast, the tuned PI controller provides a comparatively wideabsorption band with less than 80% reflection between about 0.1 Hz and0.4 Hz. There is even a substantial damping effect remaining (|r|=0.965)at 0.6 Hz, which is three times the stick-slip frequency and close tothe second resonance frequency of the drill string.

The effective inertia J of the drilling mechanism, the characteristicimpedance ζ and the stick-slip frequency ω_(s) change the absorptionbandwidth of the frequency-reflection curve in FIG. 3. In particular,the absorption bandwidth is inversely proportional to the ratio ω_(s)J/ζ. For a drilling mechanism with a large effective inertia and/or aslender drill pipe making this ratio larger (e.g. greater than 5), theabsorption bandwidth narrows. In that case, it becomes more important toensure that the estimated stick-slip period is determined moreaccurately (if possible) so that the frequency of maximum damping is asclose as possible to the actual stick-slip frequency.

The reduction in reflection coefficient magnitude and correspondingpositive damping over the entire frequency band is very important and isachieved with only a single PI controller. This is in contrast to otheractive methods that use cascade feed-back loops in series with astandard speed controller, or that rely on some measured parameter suchas drive or string torque to provide a feedback signal to the PLC. Thefilters used in the cascade feed-back functions can be suitable fordamping the fundamental stick-slip oscillations but they can causenegative damping and instabilities at higher frequencies.

In practice, the P-term for the tuned speed controller may be determinedas follows:

$P = {\frac{\zeta}{a} = \frac{{GI}_{p}}{ca}}$where G is the shear modulus of the drill string (typical value is80×10⁹ Nm⁻²), I_(P) is the cross-sectional polar moment of inertia ofthe drill string (typical value is 12.2×10⁻⁶ m⁴) and c is the speed oftorsional waves in the drill string (typical value is 3192 ms⁻¹).

To determine the I-term in practice, there are two variables to beestimated: (a) the angular frequency ω_(s) of stick-slip oscillations,and (b) the effective inertia J of the top drive. The latter isrelatively straightforward to determine and can either be calculatedfrom theoretical values of the gear inertia, the gear ratio and themotor rotor inertia, or it can be found experimentally by running anacceleration test when the top drive 30 is disconnected from the string.A typical formula for calculating top drive inertia is:J _(TD) =J _(TD0) +n _(m) n _(gear) ² J _(MR)where J_(TD0) is top drive inertia with the motor de-coupled (typicalvalue 100 kg m⁻²), n_(gear) is the gear ratio (>1), n_(m) of activemotors (default value is 1), and J_(MR) is the rotor inertia of themotor (typical value is 18.2 kg m⁻²).

There are several ways that the angular frequency ω_(s) may beestimated, including: (i) calculations from string geometry, (ii) bymanual measurement (e.g. using a stop watch) and (iii) by automaticdetermination in the PLC software. An important advantage of the PItuning aspect of embodiments of the invention is that the damping effectof stick-slip oscillations is still obtained even if the estimate of thestick-slip period used to tune the PI controller is not very accurate.For example, FIG. 3 shows maximum damping occurring at a frequency of0.2 Hz. Even if the real stick-slip frequency is lower or higher thanthis, there is still a good damping effect (r 0.8) obtained betweenabout 0.09 Hz and 0.4 Hz. Accordingly, the methods used to estimatestick-slip period do not have to be particularly accurate.

(i) String Geometry

It is possible to take a theoretical approach to determine thestick-slip period using parameters of the drill-string available on-sitein the tally book. A tally book is compiled on site for each drillstring and comprises a detailed record of the properties of each sectionof drill string (e.g. OD, ID, type of pipe), a section being defined asa length (e.g. 300 m) of the same type of drill pipe.

In the following it is assumed that the drillstring 12 consists of onedrill pipe section of length 1 with a lumped bit impedance at the lowerend, represented by Z_(b). This impedance can be a pure reactive inertiaimpedance (iωJ_(b), where J_(b) is the inertia of the bottom holeassembly) or it can be a real constant representing the lumped damping(positive or negative) at the drill bit 28. The torque equations at thetop and at the bit represent the two boundary conditions. It can beshown that these two boundary conditions can be written as the followingmatrix equation.

$\begin{matrix}{{\begin{bmatrix}{\zeta + Z_{d}} & {\zeta - Z_{d}} \\{\left( {\zeta - Z_{b}} \right)e^{{- i}\;{kl}}} & {\left( {\zeta + Z_{b}} \right)e^{i\;{kl}}}\end{bmatrix} \cdot \begin{bmatrix}\Omega^{+} \\\Omega^{-}\end{bmatrix}} = \begin{bmatrix}0 \\0\end{bmatrix}} & (11)\end{matrix}$where k is the wavenumber and Z_(d) is the impedance of the drillingmechanism.

No-trivial solutions to this system of equations exist if thedeterminant of the system matrix vanishes, that is, when

$\begin{matrix}{e^{i\; 2\;{kl}} = {\frac{\left( {\zeta - Z_{d}} \right)\left( {\zeta - Z_{b}} \right)}{\left( {\zeta + Z_{d}} \right)\left( {\zeta + Z_{b}} \right)} = {r_{d}r_{b}}}} & (12)\end{matrix}$

Here reflection coefficients at the drive r_(d) and at the bottom of thedrill string r_(b) have been introduced as follows:

$r_{d} = {{\frac{\zeta - Z_{d}}{\zeta + Z_{d}}\mspace{104mu} r_{b}} = \frac{\zeta - Z_{b}}{\zeta + Z_{b}}}$

Notice that the top drive reflection coefficient r_(d)≈−1 for a stiffspeed controller (|Z_(d)|>>ζ) and the bit reflection coefficient r_(b)equals unity for a free lower end (Z_(b)=0).

The roots of equation (12) can be written as:i2kl=ln(r _(d) r _(b))=ln|r _(d) r _(b) |+i(n2π+α_(d)+α_(b))  (13)where n is a non-negative integer and α_(d) and α_(b) are the arguments(phase angles) of the complex reflection coefficients r_(d) and r_(b),respectively. The corresponding angular resonance frequencies are

$\begin{matrix}{\omega_{n} = {\left( {\alpha_{d} + \alpha_{b} + {n\; 2\pi} - {i\;\ln{{r_{d}r_{b}}}}} \right)\frac{c}{2\; l}}} & (14)\end{matrix}$Since, in general, the magnitudes and phases of the reflectioncoefficient are frequency dependent, the above equation is transcendent,without explicit analytic solutions. However, it can be solvednumerically by a PC or other computer.

The imaginary term of the above equation represents the damping of theeigenmodes. If |r_(d)r_(b)|<1 the imaginary part of the root ispositive, thus representing a normal, positive damping causing the timefactor exp(iω_(n)t) to decay with time. In contrast, if |r_(d)r_(b)|>1the damping becomes negative, causing a small amplitude to growexponentially with time.

As an example, consider a case with a completely stiff speed controller(|r_(d)|=−1 and α_(d)=π) rotating a drill string having a finite bottomhole inertia (Z_(b)=iωJ_(b), |r_(b)|=1 and α_(d)=−2 tan⁻¹(ωJ_(b)/ζ)).Then the lowest (theoretical stick-slip) frequency ω_(s) becomes:

$\begin{matrix}{\omega_{s} = {\left( {\pi - {2\;{\tan^{- 1}\left( \frac{\omega_{s}J_{b}}{\zeta} \right)}}} \right)\frac{c}{2\; l}}} & (15)\end{matrix}$With no extra bottom hole assembly inertia this expression reduces toω_(s)=πc/(2l). Notice that the resonance frequency decreases as theinertia J_(b) increases. In the extreme case when ω_(s)J_(b)>>ζ theabove formula can be rewritten as ω_(s)≈1/√{square root over (J_(b)C)}where C=l/(GI_(p)) is the static compliance of the string. This is thewell-known formula for the natural frequency of a lumped inertia andspring system.

We have found that it is useful to study the relation between lower endspeed amplitude Ω_(s)≡Ω(x=l) and the corresponding top torqueT_(s)≡T(x=0). It can be shown from the equations above that this ratiois

$\begin{matrix}{\frac{\Omega_{s}}{T_{s}} = {\frac{{r_{d}{\exp\left( {{- i}\;{kl}} \right)}} + {\exp\left( {i\;{kl}} \right)}}{\zeta\left( {r_{d} - 1} \right)} = {{{- i}\frac{\sin({kl})}{\zeta}} - \frac{\left( {1 + r_{d}} \right){\cos({kl})}}{\left( {1 - r_{d}} \right)\zeta}}}} & (16)\end{matrix}$Using the fact that characteristic impedance can be written as ζ≡kl/(ωC)the down hole speed amplitude can be expressed by

$\begin{matrix}{\Omega_{s} = {{{- \frac{\sin({kl})}{kl}}{C \cdot i}\;\omega\; T_{s}} - {\frac{\left( {1 + r_{d}} \right){\cos({kl})}}{\left( {1 - r_{d}} \right){kl}}C\;\omega\; T_{s}}}} & (17)\end{matrix}$

Notice the that the second term vanishes if the speed controller is verystiff (r≈−1) or when kl≈π/2. However if a soft speed controller is usedand there is a high inertia near the bit so that kl for the stick-slipfrequency is significantly less than π/2, then the second term may besignificant and should not be omitted.

The theory above can be generalized to strings with many sections andalso to cases with distributed damping. If a linear damping term isincluded, the generalization causes the wave number and characteristicimpedances to be complex and not purely real. If the string consists ofm uniform sections the general wave solution consists of 2 m complexspeed amplitudes, representing pairs of up and down propagating waves.Continuity of angular speed and torsion across the section boundariescan be expressed by 2(m−1) internal boundary conditions, which add tothe two end conditions in equation (11). These can be set up as ahomogeneous 2 m×2 m matrix equation. The roots of this system ofequations are those frequencies making the system matrix singular.Although it is possible to find an analytic expression for the systemdeterminant, the solutions are found numerically by a PC or othercomputer on site. IADC/SPE 15564 provides an example of one way to dothis, and its content is hereby incorporated by reference for allpurposes.

FIGS. 4A′ and 4A″ show a typical window 50 available on the driller'sconsole that enables the driller to trigger a PC to estimate a newstick-slip period based on string geometry. In particular a table 52represents the sections of the drillstring including BHA, heavy-weightdrill pipe (HWDP), and drill pipe sections 1 to 6. Available fields foreach section are: length, outer diameter and inner diameter. The drillerfirstly determines from the on-site tally book how many sections thedrill string is divided into. In this example the drill string has eightsections. For each section the driller enters figures into the threefields. A button 54 enables the driller to trigger a new stick-slipperiod to be estimated based on the string geometry entered in the table52. In particular, the table establishes the 2 m×2 m matrix equationmentioned above and the PL (not shown) uses a numeric method to find theroots of the matrix that make the matrix singular. The smallest root isthe stick-slip period output 56 in the window 50.

(ii) Manual Estimation

To determine the stick-slip period manually, the driller may observe thedrive torque as displayed on the driller's console 34 and determine theperiod by measuring the period of the variation of the drive torque witha stopwatch. This is readily done since each period is typically 2 s to10 s. An alternative method is for the driller to listen to the changein pitch of the top drive motor and to time the period that way. Asmentioned above, such methods should be sufficient as the estimatedstick-slip frequency does not have to be particularly close to the realstick-slip frequency in order that the stick-slip oscillations aredamped.

(iii) Automatic Estimation

Automatic estimation means that the PLC software estimates thestick-slip period or frequency from measurements made during drilling.In particular, the top drive torque signal is filtered by a band-passfilter that passes frequencies in the range 0.1 Hz to 0.5 Hz (i.e. aperiod of between 2 s and 10 s), that is the filter favours thestick-slip component and suppresses all other frequency components. ThePLC then detects the period between every new zero up-crossing of thefiltered torque signal and uses these values in a recursive smoothingfilter to obtain a stable and accurate period estimate. The finalsmoothing filter is frozen when either the stick-slip severity (seebelow) falls below a low critical value, or the tuning method isactivated.

To help the period estimator to quickly find the accurate period, theoperator can either put in a realistic starting value or pick atheoretical value calculated for the actual string (determined as perString Geometry section above).

In use, the tuned PI controller is activated when there is a significantstick-slip motion (as determined by the driller or by software).However, the stick-slip frequency estimation (period measurement) takesplace before the tuned PI controller is actually used to control thedrilling mechanism. Once complete the period estimator is turned offwhen PI controller is on, because the natural period of the stick-sliposcillations can change slightly when soft speed control is used.

There does not appear to be a need for very frequent retuning of theestimated frequency because the natural stick-slip frequency variesslowly with drill string length. It is a good idea, however, toautomatically update the period at every connection i.e. when another 30m of drill pipes are added to the drill string. To do that it ispossible to use theoretical sensitivity analysis to predict how thestick-slip period increases with drill string length. One way to do this(but not the only way) is to find the theoretical periods for two stringlengths (L and L+200 m, say) and then use interpolation for the increasecaused by the addition of a 30 m section in order to update theestimated period.

Estimation of Stick-Slip Severity and Instantaneous Bit Speed

An additional aspect of some embodiments of the invention is provided asa set of computer executable instructions in the PLC software thatenables quantification of bit speed variations and an estimate of theinstantaneous bit rotation speed. ‘Bit speed’ means the BHA rotationspeed excluding the contribution from an optional mud motor. This aspectmay be provided separately from or in combination with the PI controllertuning.

This estimation is achieved by combining the known torsional complianceC of the drill string and the variations of the drive torque. Ingeneral, since the torque is not a strictly periodic signal but oftenpossesses a wide range frequencies, an accurate calculation is extremelycomplicated and is therefore not suitable for implementation in a PLC.However, we have realised that since the stick-slip motion is dominatedby the fundamental stick-slip frequency, it is possible to achievefairly good estimates based on this frequency only.

The key equation is (17) above, which describes a good approximation forthe complex speed amplitude as a function of the top string torque. Thetwo terms in this expression must be treated differently because theyrepresent harmonic components having a 90 degrees phase difference.While the imaginary factor iωT_(s) should be treated as the timederivative of the band pass filtered torque, the real term factor ωT_(s)can be approximated as the product of the band pass filtered torque andthe stick-slip frequency. Since the band pass filter suppresses allfrequencies except the stick slip-frequency, it is possible tosubstitute direct time integration by an integration basedapproximation. This approximation is based on the fact that iω≈−ω_(s)²/(iω), where 1/(iω) represents time integration. This approximationfavours the stick-slip frequency and suppresses higher harmonics. Thetime domain versions of (17) suitable for implementation in the PLC 38is:

$\begin{matrix}{\Omega_{b} = {{{{- \frac{\sin({kl})}{kl}}{C \cdot \frac{d\; T_{bp}}{d\; t}}} - {\frac{\left( {1 + r_{d}} \right){\cos({kl})}}{\left( {1 - r_{d}} \right){kl}}C\;\omega_{s}T_{bp}}} \approx {\frac{\sin({kl})}{kl}{C \cdot \omega_{s}^{2}}{\int{T_{bp}d\; t}}}}} & (18)\end{matrix}$Here the phase parameter kl=ω_(s)l/c. In the last approximation theintegral approximation for time derivation is used and the second termis omitted.

Even though the formula above is based on a single section string,simulations have shown that it also provides good estimates formulti-section strings if the total string compliance C is used:

$\begin{matrix}{C = {\sum\limits_{j = 1}^{m}\;\frac{l_{j}}{I_{p.j}G}}} & (19)\end{matrix}$

A version of the algorithm implemented in the PLC 38 to estimate bothinstantaneous BHA speed and a stick-slip severity, comprises thefollowing steps.

1. Estimate the sting torque by correcting for inertia effects (subtractthe effective motor inertia times the angular acceleration) and by usingthe gear ratio to scale it properly;

2. Band pass filter the estimated torque with a band pass filter centredat the observed/estimated stick-slip frequency. The filter should be of2nd order or higher, but can preferably be implemented in the PLC as aseries of 1st order recursive IIR filters;

3. Calculate the total static drill string compliance using equation(19) above;

4. Calculate the phase parameter kl=ω_(s)l/c where ω_(s) is thedetermined angular stick-slip frequency;

5. Calculate the dynamic downhole speed by using either the accurate orthe approximate version of equation (18) above;

6. Calculate the “stick-slip severity” σ, which is the normalizedstick-slip amplitude, determined as the ratio of dynamic downhole speedamplitude over the mean top drive rotational speed;

7. Find the instant speed as the sum of the low pass filtered top drivespeed and the estimated dynamic downhole speed. Clip to zero if theestimated speed goes negative;

8. Output data to be plotted on a graph (e.g. RPM versus time);

9. Repeat steps 1 to 8 to provide substantially real-time estimate ofbit speed.

It is envisaged that this method could be performed where only the BHAspeed estimate is output or only the stick-slip severity is output.

Regarding step 6, a possible way of estimating the stick-slip severityis to use the following formula where LP( ) denotes low pass filtering:

$\begin{matrix}{\sigma = \frac{\sqrt{2 \cdot {{LP}\left( \Omega_{b}^{2} \right)}}}{\Omega_{set}}} & (20)\end{matrix}$

Because the above method takes the reflection coefficient into account,it applies both for a standard and tuned speed control. Duringacceleration transients when the top drive speed is changedsignificantly the estimator is not reliable but can give large errors.Nonetheless we believe this is a useful tool for assessing downholeconditions, either automatically in software or by display for analysisby a driller.

The ratio of dynamic speed amplitude to the average top drive speed is adirect and quantitative measurement of the stick-slip motion, moresuitable than either the dynamic torque or the relative torqueamplitude. Even though the estimated bit speed is not highly accurate,it provides a valuable input to the driller, and monitoring of it in atrend plot will give the operator more explicit information on what ishappening at the bit.

User Interface

A user interface is provided for the driller's console 34 that comprisesa graphical interface (see FIGS. 4A′ and 4A″, and 4B′ and 4B″) whichprovides the operator with direct information on the stick-slip status.Stick-slip is indicated by three different indicators:

A “traffic light” indicator 58 in FIG. 4A′ with 3 levels of stick-slip:a green light for small amplitudes (0-30%), a yellow warning light ifthe speed oscillations are significant (30-70%) and finally a red lightif even higher amplitudes are estimated. This percentage value is basedon the stick-slip severity as determined above.

The stick-slip severity is plotted in a plot 62 of torque versus time inFIG. 4B′-4B″ to see how the stick-slip has developed over a specifiedperiod of time.

The instant bit speed estimate in a plot 64 of instantaneous bit speedversus time in FIG. 4B′-4B″ giving a visual and direct impression of thedown hole stick-slip status.

As mentioned above, the window 50 requires the operator to input a roughdescription of the string, in terms of a simplified tally. This tallyaccepts up to 8 different sections where the length, outer diameter andmass per unit length are specified. This information is used forcalculating both the theoretical estimated frequency for the lowest modeand the static drill string compliance at this frequency.

The operator can switch the tuned PI controller on or off. In the offstate, the standard drive speed controller is used. When the tuning isturned on, this speed controller is bypassed by the tuned PI controller42 which is implemented in the PLC 38. If the drive controller in thetop drive 30 is a modern digital one, it is also possible to changedrive speed controller itself, instead of bypassing it. However, if thebypass method is chosen, this is achieved by sending a high speedcommand from the PLC 38 to the speed controller in the top drive 30 andby controlling the output torque limit dynamically. In normal drillingthis torque limit is used as a safety limit preventing damage to thestring if the string suddenly sticks. In the tuned control mode, whenthe PLC 38 controls the torque limit dynamically, this limit issubstituted by a corresponding software limit in the PLC 38.

The operator can also change the prevention or mobility factor a withinpreset limits via buttons 60, typically between 0.05 and 0.33. A highfactor implies a softer speed control and less probability for thestick-slip motion to start or persist. The disadvantage of a high factoris larger fluctuations of the top drive speed in response to harmlesschanges in the string torque level. It may be necessary to choose a highfactor to cure severe stick-slip oscillations but the operator shouldreduce the factor when smooth drilling is restored.

It is envisaged that the decision to activate and de-activate the tunedspeed control may be taken by the PLC 38 or other electronic controller.Such a controller may monitor the instantaneous estimate of bit speed asset out above. When a period pattern of stick-slip is observed, thecontroller may activate the tuning. Furthermore the controller maygradually increase the mobility or prevention factor to increase thesoftness of the top drive 30 if the stick-slip oscillations do notreduce in magnitude over a predetermined period e.g. 2 minutes. Once thestick-slip oscillations have reduced or substantially disappeared thecontroller may gradually reduce the mobility factor (e.g. down to a=0.1)to improve drilling efficiency.

HIL Testing

The PI tuning method has recently been extensively tested in so-calledHardware In the Loop (HIL) simulations. In these tests the PLC programsare run on a physical PLC interfacing to a real-time simulation model ofthe drive and the drill string.

The simulation model being used for the HIL testing of tuning method hasthe following features:

1. The drive is modelled as a standard PI speed controller with torqueand power limitations and anti-windup. The torque or current controlleris perfect in the sense that the actual torque is assumed to match theset torque with no delay.

2. The model can handle a plurality of drive motors connected to theoutput shaft by a gear.

3. The drill string is modelled as a series of lumped inertia and springelements derived from any tally book. The grid length used in mostexamples below is approximately 28 m, which is the typical length of atriple stand. Hence the 3200 m long string used below consists of 114elements.4. The static friction torque is calculated for every element, based onthe theoretical contact force being a function of weight andinclination, curvature and tension. The effect of WOB and bit torque isalso included.5. The dynamic, speed dependent friction torque is modelled as a sum ofthree terms. The first term is a soft-sign variant of the Columnfriction, the second represents an extra static start friction, and thethird is a linear damping term, independent of the contact force. Tosimulate instability with growing oscillation amplitude from smoothdrilling, this damping coefficient must be negative.

The model was first developed as a Simulink model under the Matlabenvironment. It is later implemented with the Simulation Module toolboxunder the National Instrument LabView environment and run on a powerfulPC platform. Although this PC is not using a real time (RT) operativesystem, its high power makes the model RT for all practical purposes.

The LabView simulation program is linked to the PLC by a so-calledSimbaPro PCI profibus DP (Distributed Peripherals) card, which cansimulate all DP nodes connected to the PLC. The update time is set to 10ms (100 Hz), which is within the PLC cycle time (typically 20 ms).

Results from the HIL testing are shown in FIG. 5. The string used is3200 m in length similar to the string used in the field test (seebelow). The theoretical period for the lowest mode is 5.2 s. FIG. 5shows a graph 70 of the torque and speed for the drillstring (trace 72)and for the top drive (trace 74) during a 150 s period including a 5 sinterval where the top drive speed is accelerated from zero to 100 rpm.The tuned speed control is turned on 30 s after start of rotation.Steady stick-slip oscillations are established soon after the start up.The stick-slip period stabilizes around 5.3 s. This is slightly longerthan the theoretical pendulum period, but the extended period isconsistent with the fact that the sticking interval is substantial. Notethat the top drive speed is nearly constant during this part of thespeed control.

When the tuned speed control is turned on, the top drive speed (trace78) temporarily shows a pronounced dynamic variation 79 in response tothe large torque variations. But after a few periods the stick-slipmotion fades away and the top drive speed, as well as the bit speed,become smooth. When tuned speed control is turned off again, thedown-hole speed (trace 76) amplitude starts to grow, until fullstick-slip motion is developed. This instability is a consequence of thenegative damping included in the string torque model.

FIG. 6 shows results 80 from the same simulations, but now with focus onthe PLC estimated stick-slip severity (trace 87) and instantaneous bitspeed (trace 84)—note that the lower graph is a continuation of theupper graph and shows the difference between simulated speed (trace 84)and estimated speed (trace 86). The bit speed estimate is fairly goodduring steady conditions but has significant error during start-up.Despite this, the estimated bit speed is able to provide the drillerwith a useful picture of down hole speed variations. The effectivenessof the tuned speed controller is clearly illustrated by the trace 87 ofstick-slip severity: when tuned speed control is in use, the stick-slipseverity falls almost to zero. Once tuned control is switched off, thestick-slip severity increases once again.

Field Test

The tuning has been tested in the field, while drilling a long deviatedwell. The string was approximately 3200 m long with a 5.5 inch drillpipe. Unfortunately, the test ended after a relative short period ofsevere stick-slip conditions, when the PDC bit drilled into a softerformation. The new formation made the bit less aggressive with lessnegative damping, thus removing the main source of the stick-sliposcillations.

FIG. 7 shows an example where stick-slip motion is developed whilerotating with the standard stiff speed controller. Two graphs 90 areshown: one of drive torque versus time, and the other of bit speedversus time. A few comments on these graphs are given below:

The data was recorded from the PLC at a sampling rate of approximately 9Hz.

The “TD corrected” torque (trace 92) is the estimated string torque andequal to the measured drive torque corrected for inertia effects.

The TD corrected torque as well as the bit speed are estimated by postprocessing the recorded data using the methods described above.

The standard top drive speed controller is very stiff, becausevariations of the measured speed (trace 94) can barely be seen afterturning off the tuned speed control and the top drive rpm is virtuallyconstant. The corresponding small accelerations are the reason why themeasured drive torque almost matches the inertia corrected string torqueduring this period.

The high frequency torque oscillations (at 1.1 Hz) seen during firstpart of the trace 96 when tuning is on probably come from a higher moderesonance in the drill string. These vibrations seem to be independentof the type of speed controller used, but they vanish when stick-slip isdeveloped.

The prevention factor (line 98) is the operator set mobility factor amentioned above.

The observed stick-slip period is approximately 5.2 s, which is in goodagreement with the theoretical period for this particular string.

Another example of successful curing of stick-slip motion is shown inFIG. 8. In this figure a similar graph 100 to graph 90 is shown:

The “TD set” speed (trace 102) is the speed command sent to the drive.When the tuning is turned on, this level is raised so the bypassed drivespeed controller always tries to increase the torque beyond the dynamiclimit of the new speed controller. In this case the speed increase is aslightly too small, causing the dynamic speed to be clipped by the drivespeed controller. This clipping will reduce the damping effect under thetuned PI controller.

When tuning is turned on, the mobility factor (line 104) isapproximately 15%. This is a little too low, because stick-sliposcillations are not cured before the operator increases this factor at106.

After the stick-slip motion has faded at about 4310 s, the 1.1 Hzoscillations reappear with an amplitude similar to what was observedbefore. But now the vibrations are seen also in the measured speed.

Additional data, not included here, show that the 1.1 Hz oscillationamplitudes decrease but do no vanish completely when the mobility factoris further increased. It means that even though the top drive impedanceis inertia dominated at this frequency the soft PI controller also hassome dampening effect on higher mode oscillations as well.

In summary, there is described a PI controller tuning method forinhibiting detrimental stick-slip oscillations. The system comprises aPI type drive speed controller being tuned so that it effectivelydampens torsional oscillations at or near the stick-slip frequency. Itis passive in the sense that it does not require measurement of stringtorque, drive torque or currents, as alternative systems do. The dampingcharacteristics of a tuned drilling mechanism drops as the frequencymoves away from the stick-slip frequency, but the damping never dropsbelow zero, meaning that the drilling mechanism will never amplifytorsional vibrations of higher modes. The method is suitable forimplementation in the PLC controlling a drilling mechanism. The tunedPI-controller can either be implemented in the PLC itself or,alternatively, calculate the speed controller constants P and I and passto the inherent digital speed controller of the top drive motors.Various embodiments of the invention also include other useful aspects,including a screen based user interface, automatic determination of thestick-slip frequency, estimation of instantaneous bit speed andcalculation of a stick-slip severity. The latter two are based on thedrill string geometry and the measured torque signal.

In conclusion, therefore, it is seen that the embodiments of theinvention disclosed herein and those covered by the appended claims arewell adapted to carry out the objectives and obtain the ends set forth.Certain changes can be made in the subject matter without departing fromthe spirit and the scope of this disclosure. It is realized that changesare possible within the scope of this disclosure and it is furtherintended that each element or step recited in any of the followingclaims is to be understood as referring to the step literally and/or toall equivalent elements or steps. The following claims are intended tocover the disclosed principles and embodiments of the invention asbroadly as legally possible in whatever form it may be utilized. Theinvention claimed herein is new and novel in accordance with 35 U.S.C. §102 and satisfies the conditions for patentability in § 102. Theinvention claimed herein is not obvious in accordance with 35 U.S.C. §103 and satisfies the conditions for patentability in § 103. Thisspecification and the claims that follow are in accordance with all ofthe requirements of 35 U.S.C. § 112. The inventors may rely on theDoctrine of Equivalents to determine and assess the scope of theirinvention and of the claims that follow as they may pertain to apparatusnot materially departing from, but outside of, the literal scope of theinvention as set forth in the following claims. All patents, patentapplications and scientific papers identified herein are incorporatedfully herein for all purposes.

What is claimed is:
 1. A method of damping stick-slip oscillations in adrill string, the method comprising: damping said stick-sliposcillations using a drilling mechanism at a top of said drill string,wherein said stick-slip oscillations comprise torsional wavespropagating along said drill string, the drill string being atransmission line for the torsional waves; tuning a PI controller sothat said drilling mechanism absorbs torsional energy from said drillstring at a frequency that is at a frequency of said stick-sliposcillations; wherein the tuning comprises calculating an I-term of saidPI controller which uses a value of an estimated frequency of said stickslip oscillations and a value of an effective inertia of said drillingmechanism, and which step of calculating the I-term does not use a valueof the length of said drill string and a value of a speed of thetorsional waves, whereby said drilling mechanism has a frequencydependent reflection coefficient of said torsional waves, whichreflection coefficient is at a minimum at or near said frequency ofstick-slip oscillations, the minimum having a value between 0.5 (50%)and 0.9 (90%); and controlling speed of rotation of said drillingmechanism using the PI controller based upon said tuning, wherein thevalue of the estimated frequency of said stick-slip oscillations isdetermined by automatic measurement.
 2. The method of claim 1, furthercomprising adjusting said I-term according to I=ω_(s) ²J where ω_(s) anestimated angular frequency of said stick-slip oscillations and J is theeffective inertia of said drilling mechanism.
 3. The method of claim 1,further comprising adjusting said P-term such that said reflectioncoefficient does not vanish whereby a fundamental mode of said stickslip oscillations is inhibited from splitting into two new modes withdifferent frequencies.
 4. The method of claim 1, further comprisingadjusting said P-term as P=ζ/a where a is a mobility factor that permitsadjustment of said P-term during drilling, whereby energy absorption ofsaid stick-slip oscillations by said drilling mechanism is increased orreduced.
 5. The method of claim 4, further comprising increasing saidmobility factor based on the magnitude of said stick-slip oscillationsbeing not reduced.
 6. The method of claim 4, further comprising reducingsaid mobility factor once the magnitude of said stick-slip oscillationshas been reduced, whereby drilling efficiency is increased withoutre-appearance or increase in magnitude of said stick-slip oscillations.7. The method of claim 1, wherein said PI controller is separate from adrilling mechanism speed controller, the method further comprisingbypassing said drilling mechanism speed controller with said PIcontroller during damping of said stick-slip oscillations.
 8. The methodof claim 1, wherein said drilling mechanism comprises said PIcontroller, the method further comprising tuning said PI controller whensaid stick-slip oscillations occur, and leaving said PI controlleruntuned otherwise.
 9. The method of claim 1, wherein said automaticmeasurement comprises using drill string geometry to determine saidestimated frequency of said stick-slip oscillations.
 10. The method ofclaim 9, wherein said drill string comprises a bottom hole assembly, andsaid estimated frequency of said stick-slip oscillations, ω_(s), isestimated by using a computer to solve numerically$\omega_{s} = {\left( {\pi - {2\;{\tan^{- 1}\left( \frac{\omega_{s}J_{b}}{\zeta} \right)}}} \right)\frac{c}{2\; l}}$where J_(b) is the inertia of the bottom hole assembly, ζ is thecharacteristic impedance of the drill string, c is the speed oftorsional waves in the drill string, and l is the length of the drillstring.
 11. The method of claim 10, wherein said estimated frequency ofsaid stick-slip oscillations is estimated by using a computer to: dividethe drill string into m uniform sections with a lumped bit impedance atits lower end, whereby the general wave solution of a system matrix${\begin{bmatrix}{\zeta + Z_{d}} & {\zeta - Z_{d}} \\{\left( {\zeta - Z_{b}} \right)e^{- {ikl}}} & {\left( {\zeta + Z_{b}} \right)e^{ikl}}\end{bmatrix} \cdot \begin{bmatrix}\Omega^{+} \\\Omega^{-}\end{bmatrix}} = \begin{bmatrix}0 \\0\end{bmatrix}$ where Z_(d) is the impedance of the drilling mechanism,Z_(b) is the lumped bit impedance, and k is the wavenumber, comprises 2mcomplex speed amplitudes, representing pairs of up and down propagatingtorsional waves; express the continuity of angular speed and torsionacross the m uniform section boundaries by 2(m−1) internal boundaryconditions, thereby adding to the two end conditions in the equationabove; set up the 2(m−1) internal boundary conditions as a homogeneous2m×2m matrix equation; and find the roots of this system of equations asthose frequencies which make the system matrix singular, where thesmallest root is the fundamental angular frequency of stick-sliposcillations.
 12. The method of claim 1, wherein said automaticmeasurement comprises using computer observation of a drive torque ofsaid drilling mechanism to determine said estimated frequency of saidstick-slip oscillations.
 13. A method according to claim 1, wherein thelength of the drill string is up to 5000 meters.
 14. A method ofdrilling a borehole, the method comprising: rotating a drill string witha drilling mechanism so as to rotate a drill bit at a lower end of saiddrill string, the drill string being a transmission line for torsionalwaves; and in response to detection of stick-slip oscillations of saiddrill string, wherein said stick-slip oscillations comprise torsionalwaves propagating along said drill string, damping said stick-sliposcillations using a PI controller to control said drilling mechanism,the PI controller having been tuned so that said drilling mechanismabsorbs torsional energy from said drill string at a frequency that isat a frequency of said stick-slip oscillations; and tuning the PIcontroller by calculating an I-term of said PI controller using a valueof estimated frequency of said stick slip oscillations and on a value ofeffective inertia of said drilling mechanism, wherein the calculatingthe I-term does not use a value of the length of said drill string and avalue of a speed of the torsional waves, whereby said drilling mechanismhas a frequency dependent reflection coefficient of said torsionalwaves, wherein the reflection coefficient is at a minimum at or nearsaid frequency of stick-slip oscillations, the minimum having a valuebetween 0.5 (50%) and 0.9 (90%), wherein said damping includes using thePI controller to rotate the drill string based on said tuning, whereinthe value of the estimated frequency of said stick-slip oscillations isdetermined by automatic measurement.
 15. A drilling mechanism for use indrilling a borehole using a drill string, the drilling mechanismcomprising: an electronic controller having: a PI controller; and memorystoring computer executable instructions that when executed cause saidelectronic controller to: damp stick-slip oscillations using saiddrilling mechanism at a top of the drill string, wherein said stick-sliposcillations comprise torsional waves propagating along said drillstring that is a transmission line for the torsional waves; controlspeed of rotation of said drilling mechanism using said PI controllerand to: tune said PI controller so that said drilling mechanism absorbstorsional energy from said drill string at a frequency that is at afrequency of said stick-slip oscillations; and to tune said PIcontroller, calculate an I-term of said PI controller using a value ofan estimated frequency of said stick slip oscillations and a value of aneffective inertia of said drilling mechanism, and not using a value ofthe length of said drill string and a value of a speed of the torsionalwaves in the calculation of the I-term, whereby said drilling mechanismhas a frequency dependent reflection coefficient of said torsionalwaves, which reflection coefficient is at a minimum at or near saidfrequency of stick-slip oscillations, the minimum having a value between0.5 (50%) and 0.9 (90%); and determine the value of the estimatedfrequency of said stick-slip oscillations by automatic measurement,wherein the computer executable instructions, when executed, cause theelectronic controller to control the speed of rotation of the drillingmechanism based on said tuning of the PI controller.
 16. An electroniccontroller for use with a drilling mechanism for drilling a boreholeusing a drill string, the electronic controller comprising: a PIcontroller; and memory storing computer executable instructions thatwhen executed cause said electronic controller to: damp stick-sliposcillations using said drilling mechanism at a top of the drill string,wherein said stick-slip oscillations comprise torsional wavespropagating along said drill string that is a transmission line for thetorsional waves; control speed of rotation of said drilling mechanismusing said PI controller and to: tune said PI controller so that saiddrilling mechanism absorbs torsional energy from said drill string at afrequency that is at a frequency of said stick-slip oscillations; and totune said PI controller, calculate an I-term of said PI controller usinga value of an estimated frequency of said stick slip oscillations and avalue of an effective inertia of said drilling mechanism, and not usinga value of the length of said drill string and a value of a speed of thetorsional waves in the calculation of the I-term, whereby said drillingmechanism has a frequency dependent reflection coefficient of saidtorsional waves, which reflection coefficient is at a minimum at or nearsaid frequency of stick-slip oscillations, the minimum having a valuebetween 0.5 (50%) and 0.9 (90%); and determine the value of theestimated frequency of said stick-slip oscillations by automaticmeasurement, wherein the computer executable instructions, whenexecuted, cause the electronic controller to control the speed ofrotation of the drilling mechanism based on said tuning of the PIcontroller.
 17. A method of upgrading a drilling mechanism, the methodcomprising: uploading computer executable instructions to an electroniccontroller to be used on a drilling rig, wherein the electroniccontroller is for controlling operation of said drilling mechanism;wherein said computer executable instructions comprise instructions for:damping stick-slip oscillations using said drilling mechanism at a topof a drill string that is a transmission line for the torsional wave;tuning a PI controller so that said drilling mechanism absorbs torsionalenergy from said drill string at a frequency that is a frequency of saidstick-slip oscillations, wherein said stick-slip oscillations comprisetorsional waves propagating along said drill string, and wherein thetuning comprises calculating an I-term of said PI controller using avalue of an estimated frequency of said stick slip oscillations and avalue of an effective inertia of said drilling mechanism, and whereincalculating the I-term does not use a value of the length of said drillstring and a value of a speed of the torsional waves, whereby saiddrilling mechanism has a frequency dependent reflection coefficient ofsaid torsional waves, which reflection coefficient is at a minimum at ornear said frequency of stick-slip oscillations, the minimum having avalue between 0.5 (50%) and 0.9 (90%); determining the value of theestimated frequency of said stick-slip oscillations by automaticmeasurement; and controlling speed of rotation of said drillingmechanism using the PI controller based upon said tuning.
 18. The methodof claim 17, wherein said computer observation of drive torquecomprises: filtering a drilling mechanism torque signal with a band-passfilter that passes frequencies in the range 0.1 Hz to 0.5 Hz, whereby astick-slip component is passed and all other frequency components aresuppressed; using the computer to detect a period between consecutivenew zero up-crossings of the filtered torque signal; and using the zeroup-crossing values in a recursive smoothing filter to obtain an estimateof the stick-slip frequency.